Elementary functions are typically defined as a sum, product, and/or composition of finitely many polynomials, rational functions, trigonometric and exponential functions, and their inverse functions (including arcsin, log, x1/n).
The elementary functions (of x) include:
- Constant functions: etc.
- Powers of : etc.
- Roots of etc.
- Exponential functions:
- Trigonometric functions: etc.
- Inverse trigonometric functions: etc.
- Hyperbolic functions: etc.
- Inverse hyperbolic functions: etc.
- All functions obtained by adding, subtracting, multiplying or dividing any of the previous functions
- All functions obtained by composing previously listed functions
Examples of elementary functions include:
- Addition, e.g. (x+1)
- Multiplication, e.g. (2x)
- Polynomial functions
It follows directly from the definition that the set of elementary functions is closed under arithmetic operations and composition. The elementary functions are closed under differentiation. They are not closed under limits and infinite sums. Importantly, the elementary functions are not closed under integration, as shown by Liouville's theorem, see Nonelementary integral. The Liouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.
The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in extensions of the algebra. By starting with the field of rational functions, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.
A differential field F is a field F0 (rational functions over the rationals Q for example) together with a derivation map u → ∂u. (Here ∂u is a new function. Sometimes the notation u′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear
and satisfies the Leibniz product rule
An element h is a constant if ∂h = 0. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.
A function u of a differential extension F[u] of a differential field F is an elementary function over F if the function u
- is algebraic over F, or
- is an exponential, that is, ∂u = u ∂a for a ∈ F, or
- is a logarithm, that is, ∂u = ∂a / a for a ∈ F.
(this is Liouville's theorem).
- Liouville, Joseph (1833a). "Premier mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 124–148.
- Liouville, Joseph (1833b). "Second mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 149–193.
- Liouville, Joseph (1833c). "Note sur la détermination des intégrales dont la valeur est algébrique". Journal für die reine und angewandte Mathematik. 10: 347–359.
- Ritt, Joseph (1950). Differential Algebra. AMS.
- Rosenlicht, Maxwell (1972). "Integration in finite terms". American Mathematical Monthly. 79 (9): 963–972. doi:10.2307/2318066. JSTOR 2318066.